Question

Given the following numbers: 25 16 61 18 15 20 15 20 24 17 19 28, derive the mean, median, mode, variance, standard deviation, skewness, kurtosis, range, minimum, maximum, sum, and count. Interpret your results. What is the empirical rule for two standard deviations of the data?

Answer 1

Answer)

Mean = sum of the observations/number of observations

Mean = (25 + 16 +...)/12 = 23.1667

To find variance we need to follow the below steps

First subtract the mean from each and every observation and then take the square and finally add them

= (25-23.1667)^2 + (16-23.1667)^2...

= 1745.6667

Now we need to divide it by n-1 to get the variance (n-1 = 12-1 = 11)

= 158.697

Variance = 158.697

We know that standard deviation is the square root of variance

So, s.d = √{158.697} = 12.5975

Median -

To find the median, first we need to arrange the data in ascending order

15, 15, 16, 17, 18, 19, 20, 20, 24, 25, 28, 61

Now the middlemost number of this arranged data is our median

Which would lie in between 6th and 7th data point

So, median is 19.5

Mode is the value which occurs most of the times

Here there are two modes, 15 and 20 (as both occured twice and rest of them occured only once)

Minimum data value is 15

Maximum data value is 61

Range is = max - min = 61-15 = 46

Emperical rule says that

95% of the data lies with in 2 standard deviation

That is in between mean - 2*s.d and mean + 2*a.d